Geographically Weighted Regression Model

Author

Federico Jose Rodriguez

Published

October 14, 2024

Modified

October 14, 2024

In this hands-on exercise, we learn to use GWR or geographically weighted regression. GWR is a technique that takes non-stationary variables and models their relationships to an outcome of interest. We use GWR to build hedonic pricing models for the resale prices of condominiums in Singapore. (from 2015)

This exercise is based on Chapter 13 of Dr Kam’s online book which can be accessed here.

Getting Started

Data Sources

To datasets will be used for this exercise:

  • 2014 Master Plan subzone boundary in shapefile format

  • 2015 condo resale prices in csv format

Installing and launching R packages

This exercise will make use of the following R packages:

  • olsrr - for building OLS (ordinary least squares) regression models and performing diagnostic tests

  • GWmodel - for calibrating geographically weighted family of models

  • tmap - for plotting cartographic quality maps

  • corrplot - for multivariate data visualization and analysis

  • sf - spatial data handling

  • tidyverse - attribute data handling

The code chunk below uses p_load() of pacman package to check if the packages are installed in the computer. It installs them first if they are not. It then loads them into R.

pacman::p_load(olsrr, corrplot, ggpubr, sf, spdep, GWmodel, tmap, tidyverse, gtsummary)

Data Import and Preparation

Geospatial data loading and preparation

The code chunk below uses st_read() of the sf package to load the geospatial data. (master plan boundaries) This data is in svy21 projected coordinate systems.

mpsz = st_read(dsn = "data/geospatial", layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `C:\drkrodriguez\ISSS626-GAA\Hands-on\Hands-On_Ex10\data\geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

As the new object does not have EPSG information, we will use the following code with st_transform() to apply the correct code of 3414.

mpsz_svy21 <- st_transform(mpsz, 3414)
mpsz_svy21 <- st_make_valid(mpsz_svy21)

We can use st_crs() to verify that the operation was successful.

st_crs(mpsz_svy21)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

We can use st_bbox() to reveal the limits of the bounding box or the extent of the sf object.

st_bbox(mpsz_svy21) #view extent
     xmin      ymin      xmax      ymax 
 2667.538 15748.721 56396.440 50256.334 

Aspatial data loading

The code chunk below uses read_csv() of readr to import the 2015 condo resale prices from the csv file.

condo_resale = read_csv("data/aspatial/Condo_resale_2015.csv")
Rows: 1436 Columns: 23
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (23): LATITUDE, LONGITUDE, POSTCODE, SELLING_PRICE, AREA_SQM, AGE, PROX_...

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

We can verify that the load is successful and get an idea of the data structure by using a function like glimpse()

glimpse(condo_resale)
Rows: 1,436
Columns: 23
$ LATITUDE             <dbl> 1.287145, 1.328698, 1.313727, 1.308563, 1.321437,…
$ LONGITUDE            <dbl> 103.7802, 103.8123, 103.7971, 103.8247, 103.9505,…
$ POSTCODE             <dbl> 118635, 288420, 267833, 258380, 467169, 466472, 3…
$ SELLING_PRICE        <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1320…
$ AREA_SQM             <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 168,…
$ AGE                  <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22, 6,…
$ PROX_CBD             <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783402…
$ PROX_CHILDCARE       <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543, 0…
$ PROX_ELDERLYCARE     <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.121…
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.410632,…
$ PROX_HAWKER_MARKET   <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969, 0…
$ PROX_KINDERGARTEN    <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076, 0…
$ PROX_MRT             <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.528…
$ PROX_PARK            <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.116…
$ PROX_PRIMARY_SCH     <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.709…
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.709…
$ PROX_SHOPPING_MALL   <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.307…
$ PROX_SUPERMARKET     <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.581…
$ PROX_BUS_STOP        <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340, 0…
$ NO_Of_UNITS          <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34, 3…
$ FAMILY_FRIENDLY      <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0…
$ FREEHOLD             <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR       <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…

We can use head() to inspect the first few (default 6) elements. We can use it for select columns/fields, as we do in the next code chunk for longitude and latitude.

head(condo_resale$LONGITUDE) #see the data in XCOORD column
[1] 103.7802 103.8123 103.7971 103.8247 103.9505 103.9386
head(condo_resale$LATITUDE) #see the data in YCOORD column
[1] 1.287145 1.328698 1.313727 1.308563 1.321437 1.314198

We can use summary() of base R to display summary statistics across columns in the same dataframe.

summary(condo_resale)
    LATITUDE       LONGITUDE        POSTCODE      SELLING_PRICE     
 Min.   :1.240   Min.   :103.7   Min.   : 18965   Min.   :  540000  
 1st Qu.:1.309   1st Qu.:103.8   1st Qu.:259849   1st Qu.: 1100000  
 Median :1.328   Median :103.8   Median :469298   Median : 1383222  
 Mean   :1.334   Mean   :103.8   Mean   :440439   Mean   : 1751211  
 3rd Qu.:1.357   3rd Qu.:103.9   3rd Qu.:589486   3rd Qu.: 1950000  
 Max.   :1.454   Max.   :104.0   Max.   :828833   Max.   :18000000  
    AREA_SQM          AGE           PROX_CBD       PROX_CHILDCARE    
 Min.   : 34.0   Min.   : 0.00   Min.   : 0.3869   Min.   :0.004927  
 1st Qu.:103.0   1st Qu.: 5.00   1st Qu.: 5.5574   1st Qu.:0.174481  
 Median :121.0   Median :11.00   Median : 9.3567   Median :0.258135  
 Mean   :136.5   Mean   :12.14   Mean   : 9.3254   Mean   :0.326313  
 3rd Qu.:156.0   3rd Qu.:18.00   3rd Qu.:12.6661   3rd Qu.:0.368293  
 Max.   :619.0   Max.   :37.00   Max.   :19.1804   Max.   :3.465726  
 PROX_ELDERLYCARE  PROX_URA_GROWTH_AREA PROX_HAWKER_MARKET PROX_KINDERGARTEN 
 Min.   :0.05451   Min.   :0.2145       Min.   :0.05182    Min.   :0.004927  
 1st Qu.:0.61254   1st Qu.:3.1643       1st Qu.:0.55245    1st Qu.:0.276345  
 Median :0.94179   Median :4.6186       Median :0.90842    Median :0.413385  
 Mean   :1.05351   Mean   :4.5981       Mean   :1.27987    Mean   :0.458903  
 3rd Qu.:1.35122   3rd Qu.:5.7550       3rd Qu.:1.68578    3rd Qu.:0.578474  
 Max.   :3.94916   Max.   :9.1554       Max.   :5.37435    Max.   :2.229045  
    PROX_MRT         PROX_PARK       PROX_PRIMARY_SCH  PROX_TOP_PRIMARY_SCH
 Min.   :0.05278   Min.   :0.02906   Min.   :0.07711   Min.   :0.07711     
 1st Qu.:0.34646   1st Qu.:0.26211   1st Qu.:0.44024   1st Qu.:1.34451     
 Median :0.57430   Median :0.39926   Median :0.63505   Median :1.88213     
 Mean   :0.67316   Mean   :0.49802   Mean   :0.75471   Mean   :2.27347     
 3rd Qu.:0.84844   3rd Qu.:0.65592   3rd Qu.:0.95104   3rd Qu.:2.90954     
 Max.   :3.48037   Max.   :2.16105   Max.   :3.92899   Max.   :6.74819     
 PROX_SHOPPING_MALL PROX_SUPERMARKET PROX_BUS_STOP       NO_Of_UNITS    
 Min.   :0.0000     Min.   :0.0000   Min.   :0.001595   Min.   :  18.0  
 1st Qu.:0.5258     1st Qu.:0.3695   1st Qu.:0.098356   1st Qu.: 188.8  
 Median :0.9357     Median :0.5687   Median :0.151710   Median : 360.0  
 Mean   :1.0455     Mean   :0.6141   Mean   :0.193974   Mean   : 409.2  
 3rd Qu.:1.3994     3rd Qu.:0.7862   3rd Qu.:0.220466   3rd Qu.: 590.0  
 Max.   :3.4774     Max.   :2.2441   Max.   :2.476639   Max.   :1703.0  
 FAMILY_FRIENDLY     FREEHOLD      LEASEHOLD_99YR  
 Min.   :0.0000   Min.   :0.0000   Min.   :0.0000  
 1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000  
 Median :0.0000   Median :0.0000   Median :0.0000  
 Mean   :0.4868   Mean   :0.4227   Mean   :0.4882  
 3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000  
 Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  

Converting aspatial dataframe into an sf object

To convert the condo_resale object into a spatial object, we can use the following code chunk that utilizes st_as_sf() from sf package. The final line of the code chunk converts the data frame from wgs84 to svy21 using the indicated crs values.

condo_resale.sf <- st_as_sf(condo_resale,
                            coords = c("LONGITUDE", "LATITUDE"),
                            crs=4326) %>%
  st_transform(crs=3414)

We can again use head() to inspect the first few elements of the new object.

head(condo_resale.sf)
Simple feature collection with 6 features and 21 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 22085.12 ymin: 29951.54 xmax: 41042.56 ymax: 34546.2
Projected CRS: SVY21 / Singapore TM
# A tibble: 6 × 22
  POSTCODE SELLING_PRICE AREA_SQM   AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
     <dbl>         <dbl>    <dbl> <dbl>    <dbl>          <dbl>            <dbl>
1   118635       3000000      309    30     7.94          0.166            2.52 
2   288420       3880000      290    32     6.61          0.280            1.93 
3   267833       3325000      248    33     6.90          0.429            0.502
4   258380       4250000      127     7     4.04          0.395            1.99 
5   467169       1400000      145    28    11.8           0.119            1.12 
6   466472       1320000      139    22    10.3           0.125            0.789
# ℹ 15 more variables: PROX_URA_GROWTH_AREA <dbl>, PROX_HAWKER_MARKET <dbl>,
#   PROX_KINDERGARTEN <dbl>, PROX_MRT <dbl>, PROX_PARK <dbl>,
#   PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
#   PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>, PROX_BUS_STOP <dbl>,
#   NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>, FREEHOLD <dbl>,
#   LEASEHOLD_99YR <dbl>, geometry <POINT [m]>

Exploratory Data Analysis (EDA)

EDA using statistical graphics

We can produce a histogram of the selling price by using the code chunk below.

ggplot(data=condo_resale.sf, aes(x=`SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  ggtitle("Distribution of Resale Selling Price") +
  labs(x = "Selling Price", y = "Records")

The figure shows a right-skewed distribution for price– that more units were sold at lower prices.

Skewed distributions are undesirable for modeling variables but can be solved through methods like log transformation. The code chunk below creates a new variable which is the log transformation of the original selling price variable. It utilizes the function log() to perform this.

condo_resale.sf <- condo_resale.sf %>%
  mutate(`LOG_SELLING_PRICE` = log(SELLING_PRICE))

We can now replot the transformed variable in a similar method using ggplot.

ggplot(data=condo_resale.sf, aes(x=`LOG_SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  ggtitle("Distribution of log of Resale Selling Price") +
  labs(x = "log(Selling Price)", y = "Records")

The new variable has less skewness compared to the original one.

Multiple histogram plots of variables

We will use ggarrange() of the ggpubr package to produce small multiple histograms or trellis plots.

The code chunk below uses ggarrange() to produce 12 small histograms arranged in columns of 4 rows.

AREA_SQM <- ggplot(data=condo_resale.sf, aes(x= `AREA_SQM`)) + 
  geom_histogram(bins=20, color="black", fill="light blue")

AGE <- ggplot(data=condo_resale.sf, aes(x= `AGE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_CBD <- ggplot(data=condo_resale.sf, aes(x= `PROX_CBD`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_CHILDCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_CHILDCARE`)) + 
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_ELDERLYCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_ELDERLYCARE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_URA_GROWTH_AREA <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_URA_GROWTH_AREA`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_HAWKER_MARKET <- ggplot(data=condo_resale.sf, aes(x= `PROX_HAWKER_MARKET`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_KINDERGARTEN <- ggplot(data=condo_resale.sf, aes(x= `PROX_KINDERGARTEN`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_MRT <- ggplot(data=condo_resale.sf, aes(x= `PROX_MRT`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_PARK <- ggplot(data=condo_resale.sf, aes(x= `PROX_PARK`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_PRIMARY_SCH <- ggplot(data=condo_resale.sf, aes(x= `PROX_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

PROX_TOP_PRIMARY_SCH <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_TOP_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

ggarrange(AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, 
          PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT,
          PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH,  
          ncol = 3, nrow = 4)

Drawing statistical point map

We can show the geospatial distribution of resale prices using the tmap package.

The code chunk below produces an interactive map (by toggling with tmap_mode("view")) of the selling price. The set.zoom.limits argument of tm_view() constrains the minimum and the maximum zoom levels. The code chunk ends by turning interactive mode off to ensure that there is no active connection.

tmap_mode("view")
tmap mode set to interactive viewing
tm_shape(mpsz_svy21)+
  tm_polygons() +
tm_shape(condo_resale.sf) +  
  tm_dots(col = "SELLING_PRICE",
          alpha = 0.6,
          style="quantile") +
  tm_view(set.zoom.limits = c(11,14))
tmap_mode("plot")
tmap mode set to plotting

Hedonic Price Modeling in R

In this section we will use lm() of base R to build hedonic pricing models.

Simple linear regression method

We build a simple linear regression model by using SELLING_PRICE as the dependent variable and then AREA_SQM as the independent variable.

condo.slr <- lm(formula=SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)

Note that lm() returns an lm object (or c(mlm, lm) for multiple responses)

The summary and output can be obtained by using summary() and anova() functions.

summary(condo.slr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3695815  -391764   -87517   258900 13503875 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -258121.1    63517.2  -4.064 5.09e-05 ***
AREA_SQM      14719.0      428.1  34.381  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 942700 on 1434 degrees of freedom
Multiple R-squared:  0.4518,    Adjusted R-squared:  0.4515 
F-statistic:  1182 on 1 and 1434 DF,  p-value: < 2.2e-16

The output includes the estimate of the best fit line based on the coefficients table displayed. In this case it is:

\[ SELLINGPRICE = -258181.1 + 1.4719 (AREA) \]

The R-squared value of 0.4518 states that the model is able to explain 45% of the values of the selling/resale price.

The p-value of less than 0.01 indicates that the regression model is a good estimator of the resale price.

To visualize the best fit line graphically, we can produce the scatterplot and then incorporate lm() function for the smoothed line in ggplot as below.

ggplot(data=condo_resale.sf,  
       aes(x=`AREA_SQM`, y=`SELLING_PRICE`)) +
  geom_point() +
  geom_smooth(method = lm) +
  ggtitle("Fl0or area vs Resale Price") +
  labs(x = "Floor Area", y = "Resale Price")
`geom_smooth()` using formula = 'y ~ x'

Multiple linear regression method

Visualizing the relationship of the independent variables

Before building a multiple LM model, it is important to ensure that the independent variables used are not highly correlated with each other. A correlation matrix is commonly used to visually inspect the relationships between these variables.

The pairs() function of R as well as other packages can be used. For this section, we will use the corrplot package.

The code chunk below uses corrplot() from that package to show the correlation coefficient between every pair of independent variable.

corrplot(cor(condo_resale[, 5:23]), diag = FALSE, order = "AOE",
         tl.pos = "td", tl.cex = 0.5, method = "number", type = "upper")

Matrix reorder, controlled by the order argument, is important to uncover hidden structures or patterns. There are four methods available: AOE, FPC, hclust and alphabet. AOE is used in the code above and uses the angular order of the eigenvectors method.

Inspecting the output above, it is clear that FREEHOLD is highly correlated with LEASE_99YEAR– so it is best to only include one of these. For our model, we will just keep the first variable.

Building a hedonic pricing model using multiple linear regression method

The code chunk below uses lm() to calbrate a multiple linear regression model. It also produces the summary of the model using summary()

condo.mlr <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE    + 
                  PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                  PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + PROX_KINDERGARTEN + 
                  PROX_MRT  + PROX_PARK + PROX_PRIMARY_SCH + 
                  PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
                  PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                data=condo_resale.sf)
summary(condo.mlr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + 
    PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + 
    PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
    PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3475964  -293923   -23069   241043 12260381 

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)           481728.40  121441.01   3.967 7.65e-05 ***
AREA_SQM               12708.32     369.59  34.385  < 2e-16 ***
AGE                   -24440.82    2763.16  -8.845  < 2e-16 ***
PROX_CBD              -78669.78    6768.97 -11.622  < 2e-16 ***
PROX_CHILDCARE       -351617.91  109467.25  -3.212  0.00135 ** 
PROX_ELDERLYCARE      171029.42   42110.51   4.061 5.14e-05 ***
PROX_URA_GROWTH_AREA   38474.53   12523.57   3.072  0.00217 ** 
PROX_HAWKER_MARKET     23746.10   29299.76   0.810  0.41782    
PROX_KINDERGARTEN     147468.99   82668.87   1.784  0.07466 .  
PROX_MRT             -314599.68   57947.44  -5.429 6.66e-08 ***
PROX_PARK             563280.50   66551.68   8.464  < 2e-16 ***
PROX_PRIMARY_SCH      180186.08   65237.95   2.762  0.00582 ** 
PROX_TOP_PRIMARY_SCH    2280.04   20410.43   0.112  0.91107    
PROX_SHOPPING_MALL   -206604.06   42840.60  -4.823 1.57e-06 ***
PROX_SUPERMARKET      -44991.80   77082.64  -0.584  0.55953    
PROX_BUS_STOP         683121.35  138353.28   4.938 8.85e-07 ***
NO_Of_UNITS             -231.18      89.03  -2.597  0.00951 ** 
FAMILY_FRIENDLY       140340.77   47020.55   2.985  0.00289 ** 
FREEHOLD              359913.01   49220.22   7.312 4.38e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 755800 on 1417 degrees of freedom
Multiple R-squared:  0.6518,    Adjusted R-squared:  0.6474 
F-statistic: 147.4 on 18 and 1417 DF,  p-value: < 2.2e-16

Preparing publication quality table using olsrr

With reference to the results above, it is clear that some of the variables are not statistically significant. We revise the model to exclude such variables and then produce the summary using ols_regress().

condo.mlr1 <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                   PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                   PROX_URA_GROWTH_AREA + PROX_MRT  + PROX_PARK + 
                   PROX_PRIMARY_SCH + PROX_SHOPPING_MALL    + PROX_BUS_STOP + 
                   NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
                 data=condo_resale.sf)
ols_regress(condo.mlr1)
                                Model Summary                                 
-----------------------------------------------------------------------------
R                            0.807       RMSE                     751998.679 
R-Squared                    0.651       MSE                571471422208.591 
Adj. R-Squared               0.647       Coef. Var                    43.168 
Pred R-Squared               0.638       AIC                       42966.758 
MAE                     414819.628       SBC                       43051.072 
-----------------------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 
 AIC: Akaike Information Criteria 
 SBC: Schwarz Bayesian Criteria 

                                     ANOVA                                       
--------------------------------------------------------------------------------
                    Sum of                                                      
                   Squares          DF         Mean Square       F         Sig. 
--------------------------------------------------------------------------------
Regression    1.512586e+15          14        1.080418e+14    189.059    0.0000 
Residual      8.120609e+14        1421    571471422208.591                      
Total         2.324647e+15        1435                                          
--------------------------------------------------------------------------------

                                               Parameter Estimates                                                
-----------------------------------------------------------------------------------------------------------------
               model           Beta    Std. Error    Std. Beta       t        Sig           lower          upper 
-----------------------------------------------------------------------------------------------------------------
         (Intercept)     527633.222    108183.223                   4.877    0.000     315417.244     739849.200 
            AREA_SQM      12777.523       367.479        0.584     34.771    0.000      12056.663      13498.382 
                 AGE     -24687.739      2754.845       -0.167     -8.962    0.000     -30091.739     -19283.740 
            PROX_CBD     -77131.323      5763.125       -0.263    -13.384    0.000     -88436.469     -65826.176 
      PROX_CHILDCARE    -318472.751    107959.512       -0.084     -2.950    0.003    -530249.889    -106695.613 
    PROX_ELDERLYCARE     185575.623     39901.864        0.090      4.651    0.000     107302.737     263848.510 
PROX_URA_GROWTH_AREA      39163.254     11754.829        0.060      3.332    0.001      16104.571      62221.936 
            PROX_MRT    -294745.107     56916.367       -0.112     -5.179    0.000    -406394.234    -183095.980 
           PROX_PARK     570504.807     65507.029        0.150      8.709    0.000     442003.938     699005.677 
    PROX_PRIMARY_SCH     159856.136     60234.599        0.062      2.654    0.008      41697.849     278014.424 
  PROX_SHOPPING_MALL    -220947.251     36561.832       -0.115     -6.043    0.000    -292668.213    -149226.288 
       PROX_BUS_STOP     682482.221    134513.243        0.134      5.074    0.000     418616.359     946348.082 
         NO_Of_UNITS       -245.480        87.947       -0.053     -2.791    0.005       -418.000        -72.961 
     FAMILY_FRIENDLY     146307.576     46893.021        0.057      3.120    0.002      54320.593     238294.560 
            FREEHOLD     350599.812     48506.485        0.136      7.228    0.000     255447.802     445751.821 
-----------------------------------------------------------------------------------------------------------------

Preparing publication quality table using gtsummary

The gtsummary package provides an alternative way to produce publication-grade summaries in R.

The code chunk below uses tbl_regression() to create a formatted regression report.

gtsummary::tbl_regression(condo.mlr1, intercept = TRUE)
Characteristic Beta 95% CI1 p-value
(Intercept) 527,633 315,417, 739,849 <0.001
AREA_SQM 12,778 12,057, 13,498 <0.001
AGE -24,688 -30,092, -19,284 <0.001
PROX_CBD -77,131 -88,436, -65,826 <0.001
PROX_CHILDCARE -318,473 -530,250, -106,696 0.003
PROX_ELDERLYCARE 185,576 107,303, 263,849 <0.001
PROX_URA_GROWTH_AREA 39,163 16,105, 62,222 <0.001
PROX_MRT -294,745 -406,394, -183,096 <0.001
PROX_PARK 570,505 442,004, 699,006 <0.001
PROX_PRIMARY_SCH 159,856 41,698, 278,014 0.008
PROX_SHOPPING_MALL -220,947 -292,668, -149,226 <0.001
PROX_BUS_STOP 682,482 418,616, 946,348 <0.001
NO_Of_UNITS -245 -418, -73 0.005
FAMILY_FRIENDLY 146,308 54,321, 238,295 0.002
FREEHOLD 350,600 255,448, 445,752 <0.001
1 CI = Confidence Interval

With the gtsummary package, model statistics can also be added by appending them to the output using add_glance_table() or as a source not by using add_glance_source_note() as in the code chunk below.

tbl_regression(condo.mlr1, 
               intercept = TRUE) %>% 
  add_glance_source_note(
    label = list(sigma ~ "\U03C3"),
    include = c(r.squared, adj.r.squared, 
                AIC, statistic,
                p.value, sigma))
Characteristic Beta 95% CI1 p-value
(Intercept) 527,633 315,417, 739,849 <0.001
AREA_SQM 12,778 12,057, 13,498 <0.001
AGE -24,688 -30,092, -19,284 <0.001
PROX_CBD -77,131 -88,436, -65,826 <0.001
PROX_CHILDCARE -318,473 -530,250, -106,696 0.003
PROX_ELDERLYCARE 185,576 107,303, 263,849 <0.001
PROX_URA_GROWTH_AREA 39,163 16,105, 62,222 <0.001
PROX_MRT -294,745 -406,394, -183,096 <0.001
PROX_PARK 570,505 442,004, 699,006 <0.001
PROX_PRIMARY_SCH 159,856 41,698, 278,014 0.008
PROX_SHOPPING_MALL -220,947 -292,668, -149,226 <0.001
PROX_BUS_STOP 682,482 418,616, 946,348 <0.001
NO_Of_UNITS -245 -418, -73 0.005
FAMILY_FRIENDLY 146,308 54,321, 238,295 0.002
FREEHOLD 350,600 255,448, 445,752 <0.001
R² = 0.651; Adjusted R² = 0.647; AIC = 42,967; Statistic = 189; p-value = <0.001; σ = 755,957
1 CI = Confidence Interval

Checking for multicolllinearity

In the code chunk below, we use ols_vif_tol() of olsrr package to check for signs of multicollinearity.

ols_vif_tol(condo.mlr1)
              Variables Tolerance      VIF
1              AREA_SQM 0.8728554 1.145665
2                   AGE 0.7071275 1.414172
3              PROX_CBD 0.6356147 1.573280
4        PROX_CHILDCARE 0.3066019 3.261559
5      PROX_ELDERLYCARE 0.6598479 1.515501
6  PROX_URA_GROWTH_AREA 0.7510311 1.331503
7              PROX_MRT 0.5236090 1.909822
8             PROX_PARK 0.8279261 1.207837
9      PROX_PRIMARY_SCH 0.4524628 2.210126
10   PROX_SHOPPING_MALL 0.6738795 1.483945
11        PROX_BUS_STOP 0.3514118 2.845664
12          NO_Of_UNITS 0.6901036 1.449058
13      FAMILY_FRIENDLY 0.7244157 1.380423
14             FREEHOLD 0.6931163 1.442759

As the VIF of each of the independent variables is less than 10, we can safely assume that there is no multicollinearity in our model.

Testing for non-linearity

When performing multiple linear regression, we need to check whether the assumptions of linearity and additivity are not violated.

For linearity, we use the ols_plot_resid_fit() of olsrr package in the code chunk below.

ols_plot_resid_fit(condo.mlr1)

As the residuals / points lie around the zero line, we have confidence that the linearity assumption is not violated.

Test for normality

The code chunk below uses ols_plot_resid_hist() of olsrr package to check for normality.

ols_plot_resid_hist(condo.mlr1)

The output reveals that the residuals follow a normal distribution.

The oslrr package can also perform regular statistical tests for normality and display in a tabular format using ols_test_normality()

ols_test_normality(condo.mlr1)
Warning in ks.test.default(y, "pnorm", mean(y), sd(y)): ties should not be
present for the one-sample Kolmogorov-Smirnov test
-----------------------------------------------
       Test             Statistic       pvalue  
-----------------------------------------------
Shapiro-Wilk              0.6856         0.0000 
Kolmogorov-Smirnov        0.1366         0.0000 
Cramer-von Mises         121.0768        0.0000 
Anderson-Darling         67.9551         0.0000 
-----------------------------------------------

Testing for spatial autocorrelation

To test for spatial autocorrelation, we need to convert the resell prices sf data frame into a SpatialPointsDataFrame.

We first need to export the residuals of the regression model and save it as a dataframe.

mlr.output <- as.data.frame(condo.mlr1$residuals)

We then include this as a new field in the condo_resale.sf object by using the code chunk below

condo_resale.res.sf <- cbind(condo_resale.sf,
                             condo.mlr1$residuals) %>%
  rename(`MLR_RES` = `condo.mlr1.residuals`)

We then use the code chunk below to convert the object into SpatialPointsDataFrame format to be able to use spdep package functions on it.

condo_resale.sp <- as_Spatial(condo_resale.res.sf)
condo_resale.sp
class       : SpatialPointsDataFrame 
features    : 1436 
extent      : 14940.85, 43352.45, 24765.67, 48382.81  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 23
names       : POSTCODE, SELLING_PRICE, AREA_SQM, AGE,    PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN,    PROX_MRT,   PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH, PROX_SHOPPING_MALL, ... 
min values  :    18965,        540000,       34,   0, 0.386916393,    0.004927023,      0.054508623,          0.214539508,        0.051817113,       0.004927023, 0.052779424, 0.029064164,      0.077106132,          0.077106132,                  0, ... 
max values  :   828833,       1.8e+07,      619,  37, 19.18042832,     3.46572633,      3.949157205,           9.15540001,        5.374348075,       2.229045366,  3.48037319,  2.16104919,      3.928989144,          6.748192062,        3.477433767, ... 

The code chunk below creates an interactive map using tmap to visualize the data.

tmap_mode("view")
tmap mode set to interactive viewing
tm_shape(mpsz_svy21)+
  tmap_options(check.and.fix = TRUE) +
  tm_polygons(alpha = 0.4) +
tm_shape(condo_resale.res.sf) +  
  tm_dots(col = "MLR_RES",
          alpha = 0.6,
          style="quantile") +
  tm_view(set.zoom.limits = c(11,14))
Variable(s) "MLR_RES" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
tmap_mode("plot")
tmap mode set to plotting

The map reveals no clear signs of autocorrelation as there are no clear clusters with high or low residual values.

To verify this conclusion, we can perform Moran’s I test.

First, we generate the distance-based weight matrix by using dnearneigh() of spdep package.

nb <- dnearneigh(coordinates(condo_resale.sp), 0, 1500, longlat = FALSE)
summary(nb)
Neighbour list object:
Number of regions: 1436 
Number of nonzero links: 66266 
Percentage nonzero weights: 3.213526 
Average number of links: 46.14624 
10 disjoint connected subgraphs
Link number distribution:

  1   3   5   7   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24 
  3   3   9   4   3  15  10  19  17  45  19   5  14  29  19   6  35  45  18  47 
 25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44 
 16  43  22  26  21  11   9  23  22  13  16  25  21  37  16  18   8  21   4  12 
 45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 
  8  36  18  14  14  43  11  12   8  13  12  13   4   5   6  12  11  20  29  33 
 65  66  67  68  69  70  71  72  73  74  75  76  77  78  79  80  81  82  83  84 
 15  20  10  14  15  15  11  16  12  10   8  19  12  14   9   8   4  13  11   6 
 85  86  87  88  89  90  91  92  93  94  95  96  97  98  99 100 101 102 103 104 
  4   9   4   4   4   6   2  16   9   4   5   9   3   9   4   2   1   2   1   1 
105 106 107 108 109 110 112 116 125 
  1   5   9   2   1   3   1   1   1 
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 links

Next, we use nb2listw() to convert the neighbours into spatial weights.

nb_lw <- nb2listw(nb, style = 'W')
summary(nb_lw)
Characteristics of weights list object:
Neighbour list object:
Number of regions: 1436 
Number of nonzero links: 66266 
Percentage nonzero weights: 3.213526 
Average number of links: 46.14624 
10 disjoint connected subgraphs
Link number distribution:

  1   3   5   7   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24 
  3   3   9   4   3  15  10  19  17  45  19   5  14  29  19   6  35  45  18  47 
 25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44 
 16  43  22  26  21  11   9  23  22  13  16  25  21  37  16  18   8  21   4  12 
 45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 
  8  36  18  14  14  43  11  12   8  13  12  13   4   5   6  12  11  20  29  33 
 65  66  67  68  69  70  71  72  73  74  75  76  77  78  79  80  81  82  83  84 
 15  20  10  14  15  15  11  16  12  10   8  19  12  14   9   8   4  13  11   6 
 85  86  87  88  89  90  91  92  93  94  95  96  97  98  99 100 101 102 103 104 
  4   9   4   4   4   6   2  16   9   4   5   9   3   9   4   2   1   2   1   1 
105 106 107 108 109 110 112 116 125 
  1   5   9   2   1   3   1   1   1 
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 links

Weights style: W 
Weights constants summary:
     n      nn   S0       S1       S2
W 1436 2062096 1436 94.81916 5798.341

Next, we use lm.morantest() of spdep package to perform Moran’s I test for the residual spatial autocorrelation.

lm.morantest(condo.mlr1, nb_lw)

    Global Moran I for regression residuals

data:  
model: lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT +
PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data = condo_resale.sf)
weights: nb_lw

Moran I statistic standard deviate = 24.366, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Observed Moran I      Expectation         Variance 
    1.438876e-01    -5.487594e-03     3.758259e-05 

As the p-value is less than our level of confidence α = 0.05, we reject the hypothesis of spatial randomness. As the test statistic I is positive, we infer that the residuals exhibit clustering.

Building Hedonic Pricing Models using GWmodel

Building fixed bandwidth GWR model

Computing fixed bandwidth

In the code chunk below, we use bw.gwr() of the GWR package to determine an optimal fixed bandwidth. The adaptive="FALSE" argument value indicates that we are computing for a fixed bandwidth.

We use the approach argument to define the stopping rule which can either be "CV" or cross-validation approach, or "AICc" or AIC corrected approach. We use the former in the code chunk.

bw.fixed <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                     PROX_CHILDCARE + PROX_ELDERLYCARE  + PROX_URA_GROWTH_AREA + 
                     PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
                     PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                     FAMILY_FRIENDLY + FREEHOLD, 
                   data=condo_resale.sp, 
                   approach="CV", 
                   kernel="gaussian", 
                   adaptive=FALSE, 
                   longlat=FALSE)
Fixed bandwidth: 17660.96 CV score: 8.259118e+14 
Fixed bandwidth: 10917.26 CV score: 7.970454e+14 
Fixed bandwidth: 6749.419 CV score: 7.273273e+14 
Fixed bandwidth: 4173.553 CV score: 6.300006e+14 
Fixed bandwidth: 2581.58 CV score: 5.404958e+14 
Fixed bandwidth: 1597.687 CV score: 4.857515e+14 
Fixed bandwidth: 989.6077 CV score: 4.722431e+14 
Fixed bandwidth: 613.7939 CV score: 1.378294e+16 
Fixed bandwidth: 1221.873 CV score: 4.778717e+14 
Fixed bandwidth: 846.0596 CV score: 4.791629e+14 
Fixed bandwidth: 1078.325 CV score: 4.751406e+14 
Fixed bandwidth: 934.7772 CV score: 4.72518e+14 
Fixed bandwidth: 1023.495 CV score: 4.730305e+14 
Fixed bandwidth: 968.6643 CV score: 4.721317e+14 
Fixed bandwidth: 955.7206 CV score: 4.722072e+14 
Fixed bandwidth: 976.6639 CV score: 4.721387e+14 
Fixed bandwidth: 963.7202 CV score: 4.721484e+14 
Fixed bandwidth: 971.7199 CV score: 4.721293e+14 
Fixed bandwidth: 973.6083 CV score: 4.721309e+14 
Fixed bandwidth: 970.5527 CV score: 4.721295e+14 
Fixed bandwidth: 972.4412 CV score: 4.721296e+14 
Fixed bandwidth: 971.2741 CV score: 4.721292e+14 
Fixed bandwidth: 970.9985 CV score: 4.721293e+14 
Fixed bandwidth: 971.4443 CV score: 4.721292e+14 
Fixed bandwidth: 971.5496 CV score: 4.721293e+14 
Fixed bandwidth: 971.3793 CV score: 4.721292e+14 
Fixed bandwidth: 971.3391 CV score: 4.721292e+14 
Fixed bandwidth: 971.3143 CV score: 4.721292e+14 
Fixed bandwidth: 971.3545 CV score: 4.721292e+14 
Fixed bandwidth: 971.3296 CV score: 4.721292e+14 
Fixed bandwidth: 971.345 CV score: 4.721292e+14 
Fixed bandwidth: 971.3355 CV score: 4.721292e+14 
Fixed bandwidth: 971.3413 CV score: 4.721292e+14 
Fixed bandwidth: 971.3377 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 
Fixed bandwidth: 971.3408 CV score: 4.721292e+14 
Fixed bandwidth: 971.3403 CV score: 4.721292e+14 
Fixed bandwidth: 971.3406 CV score: 4.721292e+14 
Fixed bandwidth: 971.3404 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 

The output shows that the recommended bandwidth is 971.3405 (meters)

GWModel method - fixed bandwidth

We can calibrate the gwr model using fixed bandwidth and a gaussian kernel using the code chunk below.

gwr.fixed <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                         PROX_CHILDCARE + PROX_ELDERLYCARE  + PROX_URA_GROWTH_AREA + 
                         PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
                         PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                         FAMILY_FRIENDLY + FREEHOLD, 
                       data=condo_resale.sp, 
                       bw=bw.fixed, 
                       kernel = 'gaussian', 
                       longlat = FALSE)

The object contains the output and is in class gwrm. Calling the object displays the model output.

gwr.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-10-14 16:05:46.792674 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sp, bw = bw.fixed, kernel = "gaussian", 
    longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 971.3405 
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -3.5988e+07 -5.1998e+05  7.6780e+05  1.7412e+06
   AREA_SQM              1.0003e+03  5.2758e+03  7.4740e+03  1.2301e+04
   AGE                  -1.3475e+05 -2.0813e+04 -8.6260e+03 -3.7784e+03
   PROX_CBD             -7.7047e+07 -2.3608e+05 -8.3600e+04  3.4646e+04
   PROX_CHILDCARE       -6.0097e+06 -3.3667e+05 -9.7425e+04  2.9007e+05
   PROX_ELDERLYCARE     -3.5000e+06 -1.5970e+05  3.1971e+04  1.9577e+05
   PROX_URA_GROWTH_AREA -3.0170e+06 -8.2013e+04  7.0749e+04  2.2612e+05
   PROX_MRT             -3.5282e+06 -6.5836e+05 -1.8833e+05  3.6922e+04
   PROX_PARK            -1.2062e+06 -2.1732e+05  3.5383e+04  4.1335e+05
   PROX_PRIMARY_SCH     -2.2695e+07 -1.7066e+05  4.8472e+04  5.1555e+05
   PROX_SHOPPING_MALL   -7.2585e+06 -1.6684e+05 -1.0517e+04  1.5923e+05
   PROX_BUS_STOP        -1.4676e+06 -4.5207e+04  3.7601e+05  1.1664e+06
   NO_Of_UNITS          -1.3170e+03 -2.4822e+02 -3.0846e+01  2.5496e+02
   FAMILY_FRIENDLY      -2.2749e+06 -1.1140e+05  7.6214e+03  1.6107e+05
   FREEHOLD             -9.2067e+06  3.8073e+04  1.5169e+05  3.7528e+05
                             Max.
   Intercept            112793548
   AREA_SQM                 21575
   AGE                     434201
   PROX_CBD               2704596
   PROX_CHILDCARE         1654087
   PROX_ELDERLYCARE      38867814
   PROX_URA_GROWTH_AREA  78515730
   PROX_MRT               3124316
   PROX_PARK             18122425
   PROX_PRIMARY_SCH       4637503
   PROX_SHOPPING_MALL     1529952
   PROX_BUS_STOP         11342182
   NO_Of_UNITS              12907
   FAMILY_FRIENDLY        1720744
   FREEHOLD               6073636
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 438.3804 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 997.6196 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 42263.61 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41632.36 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 42515.71 
   Residual sum of squares: 2.53407e+14 
   R-square value:  0.8909912 
   Adjusted R-square value:  0.8430417 

   ***********************************************************************
   Program stops at: 2024-10-14 16:05:47.512285 

The report shows that the AICc of the gwr is signigicantly smaller than that of the global multiple lm (42263.61 < 42967.1)

Building adaptive bandwidth GWR model

Computing adaptive bandwidth

We again use bw.gwr() of the GWR package to determine the bandwidth. This time adaptive="TRUE" argument value indicates that we are computing for an adaptive bandwidth.

bw.adaptive <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE  + 
                        PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE    + 
                        PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                        PROX_PRIMARY_SCH + PROX_SHOPPING_MALL   + PROX_BUS_STOP + 
                        NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                      data=condo_resale.sp, 
                      approach="CV", 
                      kernel="gaussian", 
                      adaptive=TRUE, 
                      longlat=FALSE)
Adaptive bandwidth: 895 CV score: 7.952401e+14 
Adaptive bandwidth: 561 CV score: 7.667364e+14 
Adaptive bandwidth: 354 CV score: 6.953454e+14 
Adaptive bandwidth: 226 CV score: 6.15223e+14 
Adaptive bandwidth: 147 CV score: 5.674373e+14 
Adaptive bandwidth: 98 CV score: 5.426745e+14 
Adaptive bandwidth: 68 CV score: 5.168117e+14 
Adaptive bandwidth: 49 CV score: 4.859631e+14 
Adaptive bandwidth: 37 CV score: 4.646518e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 
Adaptive bandwidth: 25 CV score: 4.430816e+14 
Adaptive bandwidth: 32 CV score: 4.505602e+14 
Adaptive bandwidth: 27 CV score: 4.462172e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 

The output shows that 30 is the recommended data points to be used.

GWModel method - adaptive bandwidth

We can calibrate the gwr model using adaptive bandwidth and a gaussian kernel using the code chunk below.

gwr.adaptive <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                            PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + 
                            PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                            PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                            NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                          data=condo_resale.sp, bw=bw.adaptive, 
                          kernel = 'gaussian', 
                          adaptive=TRUE, 
                          longlat = FALSE)

The object contains the output and is in class gwrm. Calling the object displays the model output.

gwr.adaptive
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-10-14 16:05:52.861739 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sp, bw = bw.adaptive, kernel = "gaussian", 
    adaptive = TRUE, longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Adaptive bandwidth: 30 (number of nearest neighbours)
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -1.3487e+08 -2.4669e+05  7.7928e+05  1.6194e+06
   AREA_SQM              3.3188e+03  5.6285e+03  7.7825e+03  1.2738e+04
   AGE                  -9.6746e+04 -2.9288e+04 -1.4043e+04 -5.6119e+03
   PROX_CBD             -2.5330e+06 -1.6256e+05 -7.7242e+04  2.6624e+03
   PROX_CHILDCARE       -1.2790e+06 -2.0175e+05  8.7158e+03  3.7778e+05
   PROX_ELDERLYCARE     -1.6212e+06 -9.2050e+04  6.1029e+04  2.8184e+05
   PROX_URA_GROWTH_AREA -7.2686e+06 -3.0350e+04  4.5869e+04  2.4613e+05
   PROX_MRT             -4.3781e+07 -6.7282e+05 -2.2115e+05 -7.4593e+04
   PROX_PARK            -2.9020e+06 -1.6782e+05  1.1601e+05  4.6572e+05
   PROX_PRIMARY_SCH     -8.6418e+05 -1.6627e+05 -7.7853e+03  4.3222e+05
   PROX_SHOPPING_MALL   -1.8272e+06 -1.3175e+05 -1.4049e+04  1.3799e+05
   PROX_BUS_STOP        -2.0579e+06 -7.1461e+04  4.1104e+05  1.2071e+06
   NO_Of_UNITS          -2.1993e+03 -2.3685e+02 -3.4699e+01  1.1657e+02
   FAMILY_FRIENDLY      -5.9879e+05 -5.0927e+04  2.6173e+04  2.2481e+05
   FREEHOLD             -1.6340e+05  4.0765e+04  1.9023e+05  3.7960e+05
                            Max.
   Intercept            18758355
   AREA_SQM                23064
   AGE                     13303
   PROX_CBD             11346650
   PROX_CHILDCARE        2892127
   PROX_ELDERLYCARE      2465671
   PROX_URA_GROWTH_AREA  7384059
   PROX_MRT              1186242
   PROX_PARK             2588497
   PROX_PRIMARY_SCH      3381462
   PROX_SHOPPING_MALL   38038564
   PROX_BUS_STOP        12081592
   NO_Of_UNITS              1010
   FAMILY_FRIENDLY       2072414
   FREEHOLD              1813995
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 350.3088 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 1085.691 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 41982.22 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41546.74 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 41914.08 
   Residual sum of squares: 2.528227e+14 
   R-square value:  0.8912425 
   Adjusted R-square value:  0.8561185 

   ***********************************************************************
   Program stops at: 2024-10-14 16:05:53.74023 

Visualizing GWR Output

The output table includes various fields aside from the residuals and are all stored in the SpatialPointsDataFrame or SpatialPolygonsDataFrame object in an object called SDF.

Converting SDF into SF dataframe

To visualize the fields in SDF, we first convert it into an sf dataframe using the code chunks below

condo_resale.sf.adaptive <- st_as_sf(gwr.adaptive$SDF) %>%
  st_transform(crs=3414)
condo_resale.sf.adaptive.svy21 <- st_transform(condo_resale.sf.adaptive, 3414)
gwr.adaptive.output <- as.data.frame(gwr.adaptive$SDF)
condo_resale.sf.adaptive <- cbind(condo_resale.res.sf, as.matrix(gwr.adaptive.output))

We then use glimpse() to check the contents of the last object.

glimpse(condo_resale.sf.adaptive)
Rows: 1,436
Columns: 77
$ POSTCODE                <dbl> 118635, 288420, 267833, 258380, 467169, 466472…
$ SELLING_PRICE           <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ AREA_SQM                <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 1…
$ AGE                     <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22,…
$ PROX_CBD                <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783…
$ PROX_CHILDCARE          <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543…
$ PROX_ELDERLYCARE        <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.…
$ PROX_URA_GROWTH_AREA    <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.4106…
$ PROX_HAWKER_MARKET      <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969…
$ PROX_KINDERGARTEN       <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076…
$ PROX_MRT                <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.…
$ PROX_PARK               <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.…
$ PROX_PRIMARY_SCH        <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.…
$ PROX_TOP_PRIMARY_SCH    <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.…
$ PROX_SHOPPING_MALL      <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.…
$ PROX_SUPERMARKET        <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.…
$ PROX_BUS_STOP           <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340…
$ NO_Of_UNITS             <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34…
$ FAMILY_FRIENDLY         <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0…
$ FREEHOLD                <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR          <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ LOG_SELLING_PRICE       <dbl> 14.91412, 15.17135, 15.01698, 15.26243, 14.151…
$ MLR_RES                 <dbl> -1489099.55, 415494.57, 194129.69, 1088992.71,…
$ Intercept               <dbl> 2050011.67, 1633128.24, 3433608.17, 234358.91,…
$ AREA_SQM.1              <dbl> 9561.892, 16576.853, 13091.861, 20730.601, 672…
$ AGE.1                   <dbl> -9514.634, -58185.479, -26707.386, -93308.988,…
$ PROX_CBD.1              <dbl> -120681.94, -149434.22, -259397.77, 2426853.66…
$ PROX_CHILDCARE.1        <dbl> 319266.925, 441102.177, -120116.816, 480825.28…
$ PROX_ELDERLYCARE.1      <dbl> -393417.795, 325188.741, 535855.806, 314783.72…
$ PROX_URA_GROWTH_AREA.1  <dbl> -159980.203, -142290.389, -253621.206, -267929…
$ PROX_MRT.1              <dbl> -299742.96, -2510522.23, -936853.28, -2039479.…
$ PROX_PARK.1             <dbl> -172104.47, 523379.72, 209099.85, -759153.26, …
$ PROX_PRIMARY_SCH.1      <dbl> 242668.03, 1106830.66, 571462.33, 3127477.21, …
$ PROX_SHOPPING_MALL.1    <dbl> 300881.390, -87693.378, -126732.712, -29593.34…
$ PROX_BUS_STOP.1         <dbl> 1210615.44, 1843587.22, 1411924.90, 7225577.51…
$ NO_Of_UNITS.1           <dbl> 104.8290640, -288.3441183, -9.5532945, -161.35…
$ FAMILY_FRIENDLY.1       <dbl> -9075.370, 310074.664, 5949.746, 1556178.531, …
$ FREEHOLD.1              <dbl> 303955.61, 396221.27, 168821.75, 1212515.58, 3…
$ y                       <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ yhat                    <dbl> 2886531.8, 3466801.5, 3616527.2, 5435481.6, 13…
$ residual                <dbl> 113468.16, 413198.52, -291527.20, -1185481.63,…
$ CV_Score                <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ Stud_residual           <dbl> 0.38207013, 1.01433140, -0.83780678, -2.846146…
$ Intercept_SE            <dbl> 516105.5, 488083.5, 963711.4, 444185.5, 211962…
$ AREA_SQM_SE             <dbl> 823.2860, 825.2380, 988.2240, 617.4007, 1376.2…
$ AGE_SE                  <dbl> 5889.782, 6226.916, 6510.236, 6010.511, 8180.3…
$ PROX_CBD_SE             <dbl> 37411.22, 23615.06, 56103.77, 469337.41, 41064…
$ PROX_CHILDCARE_SE       <dbl> 319111.1, 299705.3, 349128.5, 304965.2, 698720…
$ PROX_ELDERLYCARE_SE     <dbl> 120633.34, 84546.69, 129687.07, 127150.69, 327…
$ PROX_URA_GROWTH_AREA_SE <dbl> 56207.39, 76956.50, 95774.60, 470762.12, 47433…
$ PROX_MRT_SE             <dbl> 185181.3, 281133.9, 275483.7, 279877.1, 363830…
$ PROX_PARK_SE            <dbl> 205499.6, 229358.7, 314124.3, 227249.4, 364580…
$ PROX_PRIMARY_SCH_SE     <dbl> 152400.7, 165150.7, 196662.6, 240878.9, 249087…
$ PROX_SHOPPING_MALL_SE   <dbl> 109268.8, 98906.8, 119913.3, 177104.1, 301032.…
$ PROX_BUS_STOP_SE        <dbl> 600668.6, 410222.1, 464156.7, 562810.8, 740922…
$ NO_Of_UNITS_SE          <dbl> 218.1258, 208.9410, 210.9828, 361.7767, 299.50…
$ FAMILY_FRIENDLY_SE      <dbl> 131474.73, 114989.07, 146607.22, 108726.62, 16…
$ FREEHOLD_SE             <dbl> 115954.0, 130110.0, 141031.5, 138239.1, 210641…
$ Intercept_TV            <dbl> 3.9720784, 3.3460017, 3.5629010, 0.5276150, 1.…
$ AREA_SQM_TV             <dbl> 11.614302, 20.087361, 13.247868, 33.577223, 4.…
$ AGE_TV                  <dbl> -1.6154474, -9.3441881, -4.1023685, -15.524301…
$ PROX_CBD_TV             <dbl> -3.22582173, -6.32792021, -4.62353528, 5.17080…
$ PROX_CHILDCARE_TV       <dbl> 1.000488185, 1.471786337, -0.344047555, 1.5766…
$ PROX_ELDERLYCARE_TV     <dbl> -3.26126929, 3.84626245, 4.13191383, 2.4756745…
$ PROX_URA_GROWTH_AREA_TV <dbl> -2.846248368, -1.848971738, -2.648105057, -5.6…
$ PROX_MRT_TV             <dbl> -1.61864578, -8.92998600, -3.40075727, -7.2870…
$ PROX_PARK_TV            <dbl> -0.83749312, 2.28192684, 0.66565951, -3.340617…
$ PROX_PRIMARY_SCH_TV     <dbl> 1.59230221, 6.70194543, 2.90580089, 12.9836104…
$ PROX_SHOPPING_MALL_TV   <dbl> 2.753588422, -0.886626400, -1.056869486, -0.16…
$ PROX_BUS_STOP_TV        <dbl> 2.0154464, 4.4941192, 3.0419145, 12.8383775, 0…
$ NO_Of_UNITS_TV          <dbl> 0.480589953, -1.380026395, -0.045279967, -0.44…
$ FAMILY_FRIENDLY_TV      <dbl> -0.06902748, 2.69655779, 0.04058290, 14.312764…
$ FREEHOLD_TV             <dbl> 2.6213469, 3.0452799, 1.1970499, 8.7711485, 1.…
$ Local_R2                <dbl> 0.8846744, 0.8899773, 0.8947007, 0.9073605, 0.…
$ coords.x1               <dbl> 22085.12, 25656.84, 23963.99, 27044.28, 41042.…
$ coords.x2               <dbl> 29951.54, 34546.20, 32890.80, 32319.77, 33743.…
$ geometry                <POINT [m]> POINT (22085.12 29951.54), POINT (25656.…

There are 77 fields that are included in the dataframe. We can use summary() to check the statistics of the yhat field as below.

summary(gwr.adaptive$SDF$yhat)
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
  171347  1102001  1385528  1751842  1982307 13887901 

Visualizing local R2

The code chunk below is used to create an interactive point symbol map based on the Local_R2 values.

tmap_mode("view")
tmap mode set to interactive viewing
tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "Local_R2",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))
tmap_mode("plot")
tmap mode set to plotting

Visualizing coefficient estimates

The code chunk below creates side-by-side interactive map of the standard error and t-value of the AREA_SQM variable.

tmap_mode("view")
tmap mode set to interactive viewing
AREA_SQM_SE <- tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_SE",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

AREA_SQM_TV <- tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_TV",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

tmap_arrange(AREA_SQM_SE, AREA_SQM_TV, 
             asp=1, ncol=2,
             sync = TRUE)
tmap_mode("plot")
tmap mode set to plotting

We can also focus on a particular region like the central region and show the R2 values using the code chunk below

tm_shape(mpsz_svy21[mpsz_svy21$REGION_N=="CENTRAL REGION", ])+
  tm_polygons()+
tm_shape(condo_resale.sf.adaptive) + 
  tm_bubbles(col = "Local_R2",
           size = 0.15,
           border.col = "gray60",
           border.lwd = 1)